X.Org provides an open source implementation of the network-transparent X Window System, as well as working on the standard itself.
The development work is being done as part of the freedesktop.org community, sponsored by the X.Org Foundation.

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The GNOME project provides two things: The GNOME desktop environment, an intuitive and attractive desktop for users, and the GNOME development platform, an extensive framework for building applications that integrate into the rest of the desktop.

KDE is a community of people dedicated to create a free and user-friendly computing experience.
KDE develops three pillars of software:
Plasma, one of the most popular desktop environments for Linux and BSD operating systems – adaptable to formfactors ranging from desktop PCs to smartphones.
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A loosely connected set of applications.
KDE Frameworks which offer all necessary means to easily build all kinds of applications upon. [Less]

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Wine is an implementation of the Windows API on top of X and Unix. It does not require Microsoft Windows, but can use native Windows DLLs if they are available. It provides both a development toolkit for porting Windows source code to Unix as well as a program loader, allowing many unmodified
... [More] Windows programs to run on x86 and x86_64 based Unixes. [Less]

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awesome is a highly configurable, next generation framework window manager for X. It is very fast, light and extensible.
It is primarly targeted at power users, developers and any people dealing with every day computing tasks and want to have fine-grained control on its graphical environment.

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Compiz is a compositing window manager that uses 3D graphics acceleration via OpenGL. It provides various new graphical effects and features on any desktop environment, including Gnome and KDE. (Short version: Wobbly windows and stuff)